# 2021 JMPSC Invitationals Problems/Problem 2

## Problem

Two quadrilaterals are drawn on the plane such that they share no sides. What is the maximum possible number of intersections of the boundaries of the two quadrilaterals?

## Solution

We find that it is possible to construct the maximal $\boxed{16}$ points, where each side of one quadrilateral intersects all four sides of the other quadrilateral.

~samrocksnature

## Solution 2

Take two concave quadrilaterals. Call two lines "somewhat parallel" if the different in their slopes is less than $\frac{1}{2}$. An arrow has approximately $4$ lines which are "somewhat" parallel, which means $2$ arrows that are $90^o$ to each other form $4 \cdot 4 = \boxed{16}$ intersections. $\linebreak$ ~Geometry285

## Solution 3

A line can intersect $4$ other non-parallel lines $4$ times. If we draw $2$ quadrilaterals with non-parallel sides, it would then be possible to get $4 \cdot 4 = \boxed{16}$ intersections.

~Mathdreams

The problems on this page are copyrighted by the Junior Mathematicians' Problem Solving Competition. 